There are several references to Canberra’s Series 35 MCA Operators Manual from sources on Full Width at Half Maximum (FWHM). It is not the easiest reference to find. I have reproduced the process the MCA uses to calculate FWHM.

Appendix D.4 FWHM (Canberra 1985, 164)

The Full Width Half Maximum (FWHM) algorithm calculates the current peak's width at half of its maximum amplitude. The calculation proceeds as follows:

The current ROI, Region of Interest, determines the search limits and the background.

The highest channel within this ROI, corrected for the background, is located using 3-point smoothed data.

Half Maximum value equals the counts in the highest channel (determined by a 3-point smooth) background corrected and divided by 2. If a divide error occurs, the algorithm will report zero.

Using data in the highest channel, proceed down each side to locate the channel whose counts fall below the designated Half Maximum value. The left and right FWHM points are the interpolated channels between the counts of the channel below the Half Maximum value and the counts of the channel above the Half Maximum value.

Full Width Half Maximum is the delta between the interpolated half maximum channels.

Note: Results may be misleading and insignificant of data is smoothed on narrow peaks to determine both the highest channel and Half Maximum values. Therefore, raw (unsmoothed) data is used for peaks of 20 channels or less.

Appendix D.7 Smooth (Canberra 1985, 166)

Smooth averages data between the left and right cursors, inclusive.

$$ y_i = \frac{y_{i-1} + 2y_i + y_{i+1}}{4} $$

where $y_i = $ the contents of channel i.

Bevington (2002, 235) derives Canberra's smoothing algorithm in “Appendix A.6 Data Smoothing.” The smoothing algorithm, developed from the binomial distribution, maintains the area under the distribution and the mean of the distribution. The width of the distribution will be wider after smoothing. Bevington gives guidance using this three-point smoothing.

Canberra differs from the classical calculation of the FWHM, where they smooth the peeks larger than 20 channels. They also interpolate left and right channels to yield a fractionally more accurate FWHM.

Reference

Bevington, Philip R., and D. Keith Robinson. 2002. Data Reduction and Error Analysis for the Physical Sciences. 3rd ed. Science/Engineering/Math. Boston: McGraw-Hill. http://www.worldcat.org/oclc/865237466.

Canberra. 1985. “Series 35 plus: Operator’s Manual: Appendix D: Computational Methods.” Canberra Industries.